Unit 2 - Algebra 1 2 - Formatted.pdf · Solving Multi Step Equations Writing Equations and Solving...

Unit 2 – Solving Equations 1

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– Unit 2 – [Solving Equations]

3x – 2 = 4

3x – 2 = 4 + 2 +2

3x = 6

3x = 6 3 3

x = 2

Add 2 to both sides Divide both sides by 3

Unit 2 – Solving Equations 2

To be a Successful Algebra class,

TIGERs will show…

#TENACITY during our practice, have…

I attempt all practice I attempt all homework I never give up when I don’t understand

#INTEGRITY as we help others with their work, maintain a…

I always check my answers I correct my work, I never just copy answers I explain answers, I never just give them

#GO-FOR-IT attitude, continually…

I write down all notes, even if I’m confused I remain positive about my goals I treat each day as a chance to reset

#ENCOURAGE each other to succeed as a team, and always…

I offer help when I understand the material I push my teammates to reach their goals I never let my teammates give up

#REACH-OUT and ask for help when we need it!

I ask my questions during homework check I ask my teammates for help during practice I attend enrichment/tutorials when I need to

Unit 2 – Solving Equations 3

Unit Calendar

MONDAY TUESDAY WEDNESDAY THURSDAY FRIDAY Sep 15 16 17 18 19

Classifying Numbers Number Properties 1-Step Equations

Evaluating Expressions

Combining Like Terms

Distributing

Solving 2 Step Equations

22 23 24 25 26

QUIZ Solving 2-Step Equations with Combining or

Distributing first.

Solving Multi Step Equations

Writing Equations and Solving

Review

TEST

29 30 Oct 1 2 3

Solving Equations with Variables on

Both Sides

Equations Review

Review 6-Weeks

TEST

Early Release Day

Essential Questions

What is the best method to solve a problem?

Why is it useful to write, transform, and/or solve equations that model situations?

What is a solution and how do I show that it is reasonable?

Unit 2 – Solving Equations 4

Critical Vocabulary

Real Numbers

Solution

Associative property

Commutative property

Distributive property

Algebra Tiles:

x2 x 1 -x2 -x -1

Unit 2 – Solving Equations 5

Classifying Numbers

Rational Numbers: Any number that ________ be written as a fraction / ratio. Irrational Numbers: Any number that ________ be written as a fraction / ratio. Whole Numbers: Any number that can be written without a __________ or a __________/__________ part.

Integers: Any number that can be written without a decimal/fractional part, but can be __________ or ___________.

Let’s place the following numbers: 4, -9, 0, ½, 5.3̅, π, 1.25, √9, √3

Rational Irrational

Integer

Whole

Unit 2 – Solving Equations 6

Number Properties

Unit 2 – Solving Equations 7

1-Step Equations

Equation Tiles Algebra Notation

x + 4 = 5

x + 4 = 5

x – 2 = 6

x – 2 = 6

4x = 8

4x = 8

𝑥

3 = 2

𝑥

3 = 2

Unit 2 – Solving Equations 8

Evaluating Expressions

For the following expressions: a = 2 b = -1 c = 5 x = -3 y = 6 z = -4

Examples:

ab – cy

2z2 + x a(xz – b)+2 z

Find the volume of a rectangular prism, V = lwh, if the length is 3, the width is 5, and the height is 12

Practice:

xy + z 3(b – x) abc + xyz

cx + za – 9

z(-4 + b) 2

√(𝑥2 + 𝑧2)

Find the perimeter of a rectangular, P = 2(l + w) if the width is 8 and the length is 9.

Find the density of an object, D = 𝑚

𝑣, if the volume is 8 and the mass is 32.

Kinetic Energy is found with the formula ½(mv2). Find the Kinetic Energy for an object with mass = 4 and velocity = 5.

Unit 2 – Solving Equations 9

Unit 2 – Solving Equations 10

Translating Words to Algebra

Sum

Combined Increased by

Plus ***Added to

***More than

Difference Decreased by

Minus Subtracted from

***Less than ***Fewer than

Times

Product Of

Twice Double/Triple

Is Are

Will be

Divided by Quotient

Per Half

Ratio

Plus the sum of … Twice the difference of … Minus the product of …

Unit 2 – Solving Equations 11

Examples:

The product of a number and 5

5x

Twice the difference of a number and 2

2(x-2)

3 more than twice a number

2x + 3

5 times the sum of a number and 3

5(x + 3)

The difference of a number and 4

x – 4

6 less than a number

x – 6

1 more than a number

x + 1

2 less than one-third of a number

1/3x – 2

3 subtracted from 4 times a number

4x – 3

Practice:

7 less than a number

Two-thirds of a number

Three times the sum of a number and 1

The quotient of a number and 10

The difference of a number and 7

3 more than a number

Twice the difference of a number and 5

9 more than twice a number

30 less than one-half of a number

17 subtracted from 4 times a number

2 more than triple a number

8 less than one-fifth of a number

Unit 2 – Solving Equations 12

Combining Like Terms

Must be the same ____________ and _____________ in order to combine.

The standard is to write the combined answer starting with largest exponents first and working down.

Examples With Algebra Tiles:

−7 + (−4)

−2𝑥 + 7𝑥 + 1

𝑥2 + 4𝑥 − 3𝑥

𝑥2 − 3𝑥 + 4 + 3x – 2

Examples Without Algebra Tiles:

−4𝑥 + 10𝑥

12𝑟 + 5 + 3𝑟 − 5

𝑥2 + 4 + 2𝑦2 − 6 + 5𝑥 − 3𝑦 + 𝑥2 − 𝑦2 − 2

7𝑥2𝑦 + 3𝑥2 − 2𝑦2 + 7𝑥2 − 5𝑥2𝑦

Unit 2 – Solving Equations 13

Application Problems:

Bob had three functions, g(x) = 2𝑥2 + 5𝑥 − 3, p(x) = 4𝑥 − 7, and s(x) = 3𝑥3 + 10. Find g(x) + p(x) + s(x).

Write an expression for the perimeter of the figure below:

Practice:

−10 + 3 + (−2)

5𝑥 + 3𝑥 + 7 − 𝑥 − 4

2𝑥2 + 5𝑥 + 6 − 3𝑥2

𝑥2 + 5𝑥 − 3 − 6𝑥 − 2𝑥2

Unit 2 – Solving Equations 14

Write an expression for the perimeter of the rectangle below:

10𝑎2 + 𝑏 + 5 − 2𝑎2

−7 + 5𝑥 − 3𝑥 + 𝑥2

−9𝑥 + 8𝑦 + 7𝑥𝑦 − 4 + 7𝑦 − 2𝑦

4𝑥 − 3𝑦 − 9𝑥 + 6𝑦 − 1

6𝑎 + 𝑏2 + 4𝑎2 − 8 − 2𝑎

𝑥2 + 5𝑥 − 5 − 9𝑥 − 6 + 4𝑦2

8𝑥2𝑦 − 𝑦2 + 4𝑥 − 6 + 3𝑦2 − 5𝑥2𝑦

7𝑥 − 40𝑦 − 6 + 4𝑥2 + 7𝑦 + 4

Unit 2 – Solving Equations 15

Distributing

Examples:

2(x – 4)

3(3 – x)

-3(x – 5)

(6 + 3y) 5

−3

4(36𝑥 + 60)

2(4x – 6) for x = -3

Unit 2 – Solving Equations 16

Practice:

6(4 – a)

5(n + 6) (3x – 5) 4

-4(3r − 8)

-2(e + 7)

−5(10x + 1)

- )( y25305

1

-(-2x + 5) 2

3(9𝑥 + 15)

−6(7 + x)

4(5z + 6) for z = -1 3(5 + 5x) for x = 2

Unit 2 – Solving Equations 17

Unit 2 – Solving Equations 18

Solving 2 Step Equations

Examples:

5x – 5 = -30

-x + 8 = 0

4x + 2y = 16 Solve for y

Unit 2 – Solving Equations 19

Practice: 6x + 1 = 37

- 10 + 3x = 14

-x + 4 = 9

- 6x - 9 = - 75

3x – 9 = 0

5x – 4 = -9

-4x + 9 = -15

-7x – 3 = -38

-21 = 8x + 11

𝑥

7+ 5 = 9

-12x + 3y = -6 Solve for y

- 4x + 4 = - 8

Unit 2 – Solving Equations 20

Solving 2 Step Equations with Distributing or Combining

Examples: -2(5x – 4) = -78

- 4x + 8 + 7x = 8

-75 = 5(-2x + 1)

x - 1 - 2x = - 5

2(l + w) = P Solve for w

Unit 2 – Solving Equations 21

Practice: 4(3x – 5) = -68

-10 = 7x + 2 – 9x

-2(x + 5) = -14

-4x + 3x – 8 = -7

6(2x + 1) = 18

45 = 5x + 5x - 5

-3(4x – 2) = 42

4x - 9 + 6x = 101

For a rectangle with a length equal to 3 more than its width, P = w + w + w + 3 + w + 3. Solve for w

Unit 2 – Solving Equations 22

Solving Multi Step Equations

Examples: 4)3(2 xx

9)1(315 x

x

−12=

5

6

8

h − 2=

4

3

V = lwh Solve for h

D = m

v Solve for v

Unit 2 – Solving Equations 23

Practice: d + 4(d + 6) = −1

1

2(12x − 10) = 31

5m − 2(m − 5) = 28

−10 + 4(3p + 10) = 18

8

b + 5=

10

5

6

𝑟 − 9=

3

2

33 = 3 + 5(y − 2)

x

−9=

2

3

I = prt Solve for r

5x+y

𝑎= 2 Solve for a

Unit 2 – Solving Equations 24

Writing Equations

Examples: Oceanside Bike Rental Shop charges twelve dollars plus eight dollars an hour for renting a bike. Alyssa paid eighty - four dollars to rent a bike. How many hours did she pay to have the bike checked out?

Sandy sold half of her comic books and then bought 8 more. She now has 11. How many did she begin with?

Benny bought five new baseball trading cards to add to his collection. The next day his dog ate half of his collection. There are now only thirty - seven cards left. How many cards did Benny start with?

Suppose it takes 48 chicken fingers to feed Mr. Young’s 4th grade class of 20 students. How many chicken fingers would be needed for 30 students?

Unit 2 – Solving Equations 25

Practice: Nancy bought a soft drink for 2 dollars and 8 candy bars. She spent a total of 26 dollars. How much did each candy bar cost?

Benny spent half of his allowance going to the movies. He washed the family car and earned 7 dollars. What is his weekly allowance if he ended with 13 dollars?

Danielle wants to purchase some clothes. She has $75. She buys one leather jacket for $39 and 3 pair of socks. Each pair of socks cost the same price. Determine the price of one pair of socks.

When ringing up a customer, a cashier needs 11 seconds to process payment as well as one second to scan each item being purchased. How many items would someone have to purchase in order for the transaction to take 37 seconds?

On Monday, 357 students went on a trip to the zoo. All 7 buses were filled and 7 students had to travel in cars. How many students were in each bus?

David read 40 pages of a book in 50 minutes. How many pages should he be able to read in 80 minutes?

Unit 2 – Solving Equations 26

Write an Expression for the Perimeter of the rectangle below:

Expression without Simplifying: _________________________ Simplify the expression by Combining Like Terms:___________

If the actual perimeter is known to be 42, write and solve the Equation for x:

Knowing x, what is the width? _______ What is the length? _______

Knowing both the width and length, what is the area? _________

Write an Expression for the Area of the rectangle below:

Expression without Simplifying: ____________ Simplify the expression by distributing: _____________

If the actual area is known to be 66, write and solve the Equation for x:

We know the width is 3, what is the length? _______

Knowing both the width and length, what is the perimeter? _________

Unit 2 – Solving Equations 27

Unit 2 – Solving Equations 28

Solving Equations with Variables on Both Sides

Examples: 3a +1 = a + 9

14 – 12x = 8x – 46

k + 5 = 20 – 2k

-5 + 7w = 8w + 12

Unit 2 – Solving Equations 29

Practice: pp 2022814

7981 zz

ww 1671239

vv 8234015

3263 xx

nn 81463

3538 uu

yy 4337

36244 xx

288 kk

Unit 2 – Solving Equations 30

Special Cases

When x = 0

Variable DOES NOT CANCEL, but the equation = 0

“x = 0”

2(3x + 4) – x = 8

No Solution

Variable CANCELS,

False Statement

“No Solution” or “NS”

2(3x + 4) – x = 5x + 1

Infinite Solutions

Variable CANCELS,

True Statement

“Infinite Solutions” or “All Real Numbers” or

“ℝ”

2(3x + 4) – x = 5x + 8

Unit 2 – Solving Equations 31

Unit 2 – Solving Equations 32

Equations Review / Extra Practice

2 Step Equations:

5x + 6 = 36

- 2x - 3 = - 11

- 3x + 3 = - 3

7x - 8 = - 29

2 Step Equations DISTRIBUTE FIRST:

6(5 - 3x) = - 114

- 2( - 6 + 7x) = 138

- 6( - 4x + 9) = 66

4( - 3x + 5) = - 100

Unit 2 – Solving Equations 33

2 Step Equations COMBINE LIKE TERMS FIRST:

5x + 8 + 2x = - 69

3 = - 2x + 6x + 7

- 5x - 4 + x = - 36

19 = 4x + 7 – 3x

2 Step Equations VARIABLES ON BOTH SIDES:

436 mm aa 913

9335 xx xx 10482

Unit 2 – Solving Equations 34